Module 24-M-AL-LAG Lie Algebras und Lie Groups

Non-official translation of the module descriptions. Only the German version is legally binding.

Students master advanced content and methods of the theory of Lie Algebras und Lie Groups in particular they can independently carry out very complex proofs in this area requiring a high level of mathematical expertise. Students can apply the basic concepts and technologies of Lie algebras and Lie groups. They are able to understand and analyse classification results and results of representation theory and to transfer the methods to new examples.
Students will be introduced to current research questions in the area of Lie Algebras und Lie Groups. They are able to recognise and assess further development opportunities and research goals.
Furthermore, students recognise further-reaching connections to mathematical issues that have already been worked out. They can transfer and apply the knowledge and methods they have learnt so far to deeper mathematical problem areas. Students also expand their mathematical intuition as a result of more intensive study.
In combination with other in-depth modules, they will be able to write their own research papers, e.g. a master's thesis in the field of the theory of Lie Algebras und Lie Groups.
In the tutorials, students develop their ability to discuss mathematical topics and thus further prepare themselves for the requirements of the Master's module, in particular for the scientific discussion within the Master's seminar presentation and the defence of their Master's thesis.

Lie groups are groups with a differentiable manifold structure such that the group operations are compatible with the manifold structure. As key tools in the description of continuous symmetries they are ubiquitous in mathematics, but also play a role in mathematical physics.

One may associate to a Lie group a Lie algebra -- explicitly constructible as the group's tangent space at the identity element --, which captures much of the "local" structure. "Exponential" and "logarithm" maps translate between Lie groups and their associated Lie algebra, facilitating a sort of transfer of several problems about Lie groups in the (linear) realm of Lie algebras.
One of the main objectives of the course will be the classification of (complex, say) simple Lie algebras in terms of Dynkin diagrams, one of the most pervasive results of modern algebra.

The content of teaching from the field of Lie Algebras und Lie Groups is:
1. Lie Algebras (nilpotent, solvable and semisimple Lie algebras, ideals, subalgebras, representations, root space decomposition, Killing form)
2. Classification of finite-dimensional semi-simple complex Lie algebras (Dynkin diagrams, Cartan matrices)
3. Lie groups and their Lie algebras (linear Lie groups, exponential and logarithm maps, Baker-Campbell-Hausdorff formula
4. Representation theory of Lie groups and algebras (selection: e.g. highest weight theory, Weyl character formula)

This module prepares the content of a master's thesis.

Linear Algebra

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